The McCulloch Neuron (1943)
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1 The McCulloch Neuron 943 p p b a pn n n g p b g t p b a [0;] p for n p p b A g tep functon The eucldan pace R n dvded n to regon A and B B p Laboratóro de Automação e Robótca - A. Bauchpe Soft Computng - Neural Netor and Fuzz Logc 5
2 The McCulloch Neuron a pattern clafer The mage cannot be dplaed. Your computer ma not have enough memor to open the mage, or the mage ma have been corrupted. Retart our computer, and then open the fle agan. If the red tll appear, ou ma have to delete the mage and then nert t agan. o o o o o o o o o o o o Lnearl eparable collecton Lnearl dependent non-eparable collecton Some Boolean functon of to varable repreented n a bnar plan. Laboratóro de Automação e Robótca - A. Bauchpe Soft Computng - Neural Netor and Fuzz Logc 5
3 Lnear and Non-Lnear Clafer There et n m poble logcal functon connectng n nput to one bnar output. n # of bnra pattern # of logcal functon # lnearl eparable % lnearl eparable , , , , , , , 0-3 The logcal functon of one varable: A, A, 0, The logcal functon of to varable: A, B, A, B, 0, A B, A B, A B, A B, A B, A B, A B, A B, A B, A B Laboratóro de Automação e Robótca - A. Bauchpe Soft Computng - Neural Netor and Fuzz Logc 53
4 To Step Bnar Perceptron The mage cannot be dplaed. Your computer ma not have enough memor to open the mage, or the mage ma have been corrupted. Retart our computer, and then open the fle agan. If the red tll appear, ou ma have to delete the mage and then nert t agan. The neuron 6 mplement a logcal AND functon b choong b For eample: ; b6 a6 f and onl f a3 a4 a5 3 Laboratóro de Automação e Robótca - A. Bauchpe Soft Computng - Neural Netor and Fuzz Logc 54
5 Three Step Bnar Perceptron A p A B p 5 0 a p B a A B ^ p 8 Laboratóro de Automação e Robótca - A. Bauchpe Soft Computng - Neural Netor and Fuzz Logc 55
6 Neuron and Artfcal Neural Netor Mcro-tructure charactertc The mage cannot be dplaed. Your computer ma not have enough of memor to each open the mage, or the mage neuron ma have been corrupted. Retart n our computer, the and then netor open the fle agan. If the red tll appear, ou ma have to delete the mage and then nert t agan. Meo-Structure organzaton of the netor Macro-Structure aocaton of netor, eventuall th ome analtcal proceng approach for comple problem p p Ba nput b p n n Ba: th p0, output 0 tll poble! Laboratóro de Automação e Robótca - A. Bauchpe Soft Computng - Neural Netor and Fuzz Logc 56
7 Lnear Tpcal actvaton functon f Hopfeld pureln BSB f Sgnal f e e 0 < 0 Perceptron hardlm f - Step f e 0 0 e < 0 Perceptron BAM hardlm f Hopfeld/ BAM e > 0 f e < 0 unchanged f 0 Hopfeld BAM f - Laboratóro de Automação e Robótca - A. Bauchpe Soft Computng - Neural Netor and Fuzz Logc 57
8 Tpcal actvaton functon BSB or Logcal Threhold f K e K e K < < K K e K BSB atln atln K f -K Logítc f e Perceptron Hopfeld BAM, BSB logg f Hperbolc Tangent e f tanh e Perceptron Hopfeld BAM, BSB tang f - Laboratóro de Automação e Robótca - A. Bauchpe Soft Computng - Neural Netor and Fuzz Logc 58
9 Meo-Structure Netor Organzaton... # neuron per laer # netor laer # connecton tpe forard, bacard, lateral. - Multlaer Feedforard Multlaer Perceptron MLP Laboratóro de Automação e Robótca - A. Bauchpe Soft Computng - Neural Netor and Fuzz Logc 59
10 Meo-Structure Netor Organzaton... - Sngle Laer laterall connected BSB elf-feedbac, Hopfeld 3 Blaer Feedforard/Feedbacard Laboratóro de Automação e Robótca - A. Bauchpe Soft Computng - Neural Netor and Fuzz Logc 60
11 Meo-Structure Netor Organzaton 4 Multlaer Cooperatve/Comparatve Netor 5 Hbrd Netor Rede Netor Sub- Sub- Rede Netor Laboratóro de Automação e Robótca - A. Bauchpe Soft Computng - Neural Netor and Fuzz Logc 6
12 Neural Macro-Structure Rede NetW. - # netor - connecton tpe - ze of netor - degree of connectvt Rede a NetW. Rede b NetW. Rede c NetW. Rede 3 NetW. Laboratóro de Automação e Robótca - A. Bauchpe Soft Computng - Neural Netor and Fuzz Logc 6
13 Delta Rule Perceptron Superved Learnng µδ δ d µ learnng rate Wdro-Hoff delta rule LMS ADALINE, MADALINE Generalzed Delta Rule d - δ µδ Wdro-Hoff Delta Rule LMS Laboratóro de Automação e Robótca - A. Bauchpe Soft Computng - Neural Netor and Fuzz Logc 63
14 Delta rule Perceptron Perceptron Roenblatt, 957 Dnamc: f b e 0 0 e < 0 p p b d - δ p p n n δ d µ δ Delta Rule µ - learnng rate δ 0 the eght not changed. Pcholog Reaonng: - potve renforcement - negatve renforcement Laboratóro de Automação e Robótca - A. Bauchpe Soft Computng - Neural Netor and Fuzz Logc 64
15 ADALINE and MADALINE Wdro & Hoff, 960 Mult. Adaptve Lnear Element p b Tranng: ε d d p b µε Wdro-Hoff delta rule LMS Leat Mean Squared algorthm 0.< µ < tablt and convergenc peed. MatLab: NEWLIN, NEWLIND, ADAPT, LEARNWH Ob : ε δ µ δ Delta Rule Laboratóro de Automação e Robótca - A. Bauchpe Soft Computng - Neural Netor and Fuzz Logc 65
16 LMS Algorthm f : R n R Obectve: learn a functon from the ample, d { }, {d } and {e } tatonar tochatc procee e d actual tochatc error Lnear neuron n t Epected value E[e ] E[d- ] E[d- t ] E[d ] E[d] t E [ t ] t Aumng determntc. Wth E [ t ] R autocorrelaton nput matr E [d] P cro correlated vector E[e ] E[d ] P t R t 0 * R P Partal dervatve equal 0 for optmal * * PR - Optmal analtc oluton of the optmzaton olveln.m Laboratóro de Automação e Robótca - A. Bauchpe Soft Computng - Neural Netor and Fuzz Logc 66
17 Iteratve LMS Algorthm R Obectve: adaptvel learn a functon from the ample, d f : R n Knong P and R, R -, then for ome : E[e ] R P Pot-multpltng b ½ R - ½ E[e ] R - P R - * * ½ E[e ] R - * c E[e ] R - c ½ Neton method LMS Hpothe: E[e e 0, e,... e ] e Ho to, cautoul fnd ne better value for, the free parameter? Laboratóro de Automação e Robótca - A. Bauchpe Soft Computng - Neural Netor and Fuzz Logc 67
18 Laboratóro de Automação e Robótca - A. Bauchpe Soft Computng - Neural Netor and Fuzz Logc Iteratve LMS Algorthm Iteratve adaptve oluton The optmal oluton never reached! MADALINE -nput, -neuron µδ * n e e,! n d d,! n d d,! n e!, [ ], t n e e! aumng R I etmated teppet decent algorthm: c e Gradent of e th repect to e LMS algorthm reduce to c e Norma- lzaton
19 The Multlaer Perceptron - The Generalzed Delta Rule Rumelhart, Hnton e Wllam, PDP/MIT, p 0 p 0 p Neuron Dnamc: Proceng Element PE n laer nput th f actvaton functon contnuou dfferentable 0 f Turnng Pont Queton: Ho to fnd the error aocated th an nternal neuron?? Laboratóro de Automação e Robótca - A. Bauchpe Soft Computng - Neural Netor and Fuzz Logc 69
20 The generalzed delta rule Tranng ε, m d,..., o m - quadratc error - egth of PE 0 p 0 p 0 p 3 3 3,,..., n - nput vector of PE Wth Intantaneou gradent: ε ε ε 0 ε, ε,! ε m o ε ε Defnng the quadratc dervatve error a δ δ ε Gradent of the error th repect to the eght a functon of the former laer gnal!! Laboratóro de Automação e Robótca - A. Bauchpe Soft Computng - Neural Netor and Fuzz Logc 70
21 The generalzed delta rule... For the output laer, the quadratc dervatve error : δ N d N d f 0 p 0 p 0 p The partal dervatve are 0 for δ d f d f d f d f ʹ The output error aocated th PE, n the lat laer: ε d d Gvng: δ ε. f ʹ Remember, actvaton functon, f, contnuou dfferentable Laboratóro de Automação e Robótca - A. Bauchpe Soft Computng - Neural Netor and Fuzz Logc 7
22 Laboratóro de Automação e Robótca - A. Bauchpe Soft Computng - Neural Netor and Fuzz Logc The generalzed delta rule... 0 p 3 0 p p 7 For a hdden laer, the quadratc dervatve error can be calculated ung the lnear output of laer : N ε ε δ N N δ ε Chan Rule N 0 Tang nto account that 0 N N l l l f δ δ N l N l l f δ δ that and f 0 conderng l f f l f ʹ. N f ʹ!! "! $! # ε δ We have: δ. f ʹ ε δ Fnall, the quadratc dervatve errror for a hdden laer:
23 The Error Bacpropagaton algorthm. random, ntalze the netor egth. for,d, tranng par, obtan. Feedforard propagaton: ε d 3. lat laer 4. for each element n the laer do: Compute ε ung ε d d f the lat laer, m ε N δ f t a hdden laer; Compute δ ε. f ʹ 5. f > 0 go to tep 4, ele contnue. 6. n n µδ 7. For the net tranng par go to tep. Laboratóro de Automação e Robótca - A. Bauchpe Soft Computng - Neural Netor and Fuzz Logc 73
24 The Bacpropagaton Algorthm n practce In the tandard form BP ver lo. BP Pathologe: paral n regon of mall gradent. 3 Intal condton can lead to local mnma. 4 Stop condton number of epoch, < ϵ e E - Energa da rede Bad Padrão Start epúro Good Start Valor Incal 5 BP varant - tranbpm th momentum - tranbp adaptve learnng rate tranlm Levenberg-Marquard J, Jacoban Δ W J J µ J T J T e Padrõe armazenado Local Mnma Padrão recuperado Optmum e, - Illutratve quadratc error a functon of the eght Ob: the error urface, normall, unnon., Etado Steepet decent go n the oppote drecton of the local gradent donhll. Laboratóro de Automação e Robótca - A. Bauchpe Soft Computng - Neural Netor and Fuzz Logc 74
25 Computatonal Tool SNNS MatLab - Neural Netor Toolbo NeuralWor Java C Hardare Implementaton of RNA Laboratóro de Automação e Robótca - A. Bauchpe Soft Computng - Neural Netor and Fuzz Logc 75
26 SNNS - Stuttgarter Neural Netor Smulator Laboratóro de Automação e Robótca - A. Bauchpe Soft Computng - Neural Netor and Fuzz Logc 76
27 MatLab - complete envronment -Stem Smulaton -Tranng -Control pureln logg tang q Model Reference Controller Reference Plant Output Neural Netor Controller Control Sgnal model lq 4 order q h4 h4 h4 Scope radba 3 fu Fcn -C- z Unt Dela ncndun nanbun Dcrete State -Space K*u Matr Gan 4 Contant 3 Stch fu Fcn u fu Fcn -C- Contant ncndun K*u z nanbun K*u Unt Dela 5 Dcrete State -Space Matr tang Gan 3 Matr netum pureln netum Gan -C- Contant 4 Stch Stch Saturaton B_c Zero -Order Hold uhat Contant 7 r fu Fcn3 ncndun nanbun Dcrete State -Space 3 K*u Matr Gan -C- Contant 5 Stch3 B_c Contant 6 Laboratóro de Automação e Robótca - A. Bauchpe Soft Computng - Neural Netor and Fuzz Logc 77
28 Demontraton - Perceptron % Perceptron % Tranng an ANN to learn to claf a non-lnear problem % Input Pattern P[ % Target %T[ 0 0 0] % Lnear eparable ] T Y T[ ] % non eparable % Tr th Roenblat' Perceptron netnepp,t,'hardlm' % tran the netor nettrannet,p,t T Y Ymnet,P Laboratóro de Automação e Robótca - A. Bauchpe Soft Computng - Neural Netor and Fuzz Logc 78
29 Demontraton - OCR Tranng Vector p p p 63 6 N 0 % Noe ANN Laboratóro de Automação e Robótca - A. Bauchpe Soft Computng - Neural Netor and Fuzz Logc 79
30 Demontraton OCR... p % of mclafcaton Neural OCR Clafer p p 63 6 N Tranng th 0 0,0,0,30,40,50 % noe No pattern ued n tranng untl % of bt flpped * - error thout no tranng pattern * - error ung no tranng pattern Wth Some No Tranng Pattern Learn ho to treat an noe Laboratóro de Automação e Robótca - A. Bauchpe Soft Computng - Neural Netor and Fuzz Logc 80
31 Demontraton LMS, ADALINE, FIR u u u! u n 0 Y z U z 0 z z! n z n n FIR Model ala table, onl Zero Ob : IIR Model more compact, but can be untable! g ec g80 50ec Stem change at 80 ec Samplng Tme, T 0.ec 0 - TDL Tme Dela Lne ec Laboratóro de Automação e Robótca - A. Bauchpe Soft Computng - Neural Netor and Fuzz Logc 8
32 Demo LMS, ADALINE, FIR... % ADALINE - Adaptve dnamc tem dentfcaton % Frt ampled tem - untl 80 ec gtf,[. ], gdcdg,. % Stem change dramatcall - after 80 ec gtf3,[ ],gdcdg,. % Peudo Random Bnar Sgnal - good for dentfcaton udnput0*0,'prbs',[0 0.0],[- ]; % tme vector... [,t,]lmgd,u,t; [,t,]lmgd,u,t,; % Create ne adalne neor th delaed nput FIR % Learnng Rate 0.09 netnelnt,,[ ],0.09 [net,y,e]adaptnet,t, % degn an average tranfer functon netdnelndt, Laboratóro de Automação e Robótca - A. Bauchpe Soft Computng - Neural Netor and Fuzz Logc 8
33 Demo LMS, ADALINE, FIR... 4 n0, lr0. RMSE Set udnput500,'prbs',[0 0.0] 0 - ADALINE Learn Stem AND alo Change n the Dnamc!! Error n0, lr0. RMSE Set.787 Verfcaton Sgnal udnput00,'prbs',[0 0.05] But, n other frequenc range not o good... need to Adut TDL, lr, T Error Laboratóro de Automação e Robótca - A. Bauchpe Soft Computng - Neural Netor and Fuzz Logc 83
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